The size and power of Student'st-test are discussed under weaker than normal conditions. It is shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of thet-statistic. (The symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.) The connection between Student's test and the usual non-parametric tests is examined, as well as power considerations involving Winsorization and permutation tests. Simultaneous use of different one-sample tests is also discussed.